let s, s1 be Real_Sequence; :: thesis: ( ( for n being Nat holds
( s . n > 0 & s1 . n = 1 / (s . n) ) ) implies for n being Nat holds (Partial_Sums s1) . n > 0 )
defpred S₁[ Nat] means (Partial_Sums s1) . $1 > 0 ;
assume A1: for n being Nat holds
( s . n > 0 & s1 . n = 1 / (s . n) ) ; :: thesis: for n being Nat holds (Partial_Sums s1) . n > 0
then A2: s1 . 0 = 1 / (s . 0) ;
A3: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A4: (Partial_Sums s1) . n > 0 ; :: thesis: S₁[n + 1]
A5: (Partial_Sums s1) . (n + 1) = ((Partial_Sums s1) . n) + (s1 . (n + 1)) by SERIES_1:def 1;
A6: s . (n + 1) > 0 by A1;
s1 . (n + 1) = 1 / (s . (n + 1)) by A1;
hence S₁[n + 1] by A4, A5, A6; :: thesis: verum

end;

s . 0 > 0 by A1;
then A7: S₁[ 0 ] by A2, SERIES_1:def 1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A7, A3);
hence for n being Nat holds (Partial_Sums s1) . n > 0 ; :: thesis: verum